偏微分方程

Chapter1TheSingleFirst-OrderEquation

1.Introduction

2.Examples

3.AnalyticSolutionandApproximationMethodsinaSimpleExample

Problems

4.Quasi-linearEquations

5.TheCauchyProblemfortheQuasi-linearEquation

6.Examples

Problems

7.TheGeneralFirst-OrderEquationforaFunctionofTwoVariables

8.TheCauchyProblem

9.SolutionsGeneratedasEnvelopes

Problems

Chapter2Second-OrderEquations:HyperbolicEquationsforFunctionsofTwoIndependentVariables

1.CharacteristicsforLinearandQuasi-linearSecond-orderEquations

2.PropagationofSingularities

3.TheLinearSecond-OrderEquation

Problems

4.TheOne-DimensionalWaveEquation

Problems

5.SystemsofFirst-OrderEquations

6.AQuasi-linearSystemandSimpleWaves

Problem

Chapter3CharacteristicManifoldsandtheCauchyProblem

1.NotationofLaurentSchwartz

Problems

2.TheCauchyProblem

Problems

3.RealAnalyticFunctionsandtheCauchy-KowalevskiTheorem

(a)Multipleinfiniteseries

Problems

(b)Realanalyticfunctions

Problems

(c)Analyticandrealanalyticfunctions

Problems

(d)TheproofoftheCauchy-Kowalevskitheorem

Problems

4.TheLagrange-GreenIdentity

5.TheUniquenessTheoremofHolmgren

Problems

6.DistributionSolutions

Problems

Chapter4TheLaplaceEquation

1.GreensIdentity.FundamentalSolutions,andPoissonsEquation

Problems

2.TheMaximumPrinciple

Problems

3.TheDirichletProblem,GreensFunction,andPoissonsFormula

Problems

4.ProofofExistenceofSolutionsfortheDirichletProblemUsingSubharmonicFunctions("PerronsMethod")

Problems

5.SolutionoftheDirichletProblembyHilbert-SpaceMethods

Problems

Chapter5HyperbolicEquationsinHigherDimensions

1.TheWaveEquationinn-DimensionalSpace

(a)Themethodofsphericalmeans

Problems

(b)Hadamardsmethodofdescent

Problems

(c)DuhamelsprincipleandthegeneralCauchyproblem

Problem

(d)Initial-boundary-valueproblems("Mixed"problems)

Problems

2.Higher-OrderHyperbolicEquationswithConstantCoefficients

(a)Standardformoftheinitial-valueproblem

Problem

(b)SolutionbyFouriertransformation

Problems

(c)SolutionofamixedproblembyFouriertransformation

(d)Themethodofplanewaves

Problems

3.SymmetricHyperbolicSystems

(a)Thebasicenergyinequality

Problems

(b)Existenceofsolutionsbythemethodoffinitedifferences

Problems

(c)Existenceofsolutionsbythemethodofapproximationbyanalyticfunctions(MethodofSchauder)

Chapter6Higher-OrderEllipticEquationswithConstantCoefficients

1.TheFundamentalSolutionforOddn

Problems

2.TheDirichletProblem

Problems

3.MoreontheHilbertSpaceH~andtheAssumptionofBoundaryValuesintheDirichletProblem

Problems

Chapter7

ParabolicEquations

1.TheHeatEquation

(a)Theinitial-valueproblem

Problems

(b)Maximumprinciple,uniqueness,andregularity

Problem

(c)Amixedproblem

Problems

(d)Non-negativesolutions

Problems

2.TheInitial-ValueProblemforGeneralSecond-OrderLinear

ParabolicEquations

(a)Themethodoffinitedifferencesandthemaximumprinciple

(b)Existenceofsolutionsoftheinitial-valueproblem

Problems

H.LewysExampleofaLinearEquationwithoutSolutions

Problems

Bibliography

Glossary

Index

　　Aconsiderableamountofnewmaterialhasbeenaddedtothisedition.ThereisanextensivediscussionofrealanalyticfunctionsofseveralvariablesinChapter3.ThispermitsestimationofthesizeofthedomainofexistenceintheCauchy-Kowalevskitheorem.AfirstapplicationoftheseestimatesconsistsinarigorousproofofanewversionofHolmgrensuniquenesstheoremforlinearanalyticpartialdifferentialequations(onlysketchedintheearliereditions).